Specifically, I'm looking for what areas of science would need to have been developed for the manufacturing process of jet engines to be commonplace for large manufacturing firms.
For context, my world is relatively low-tech with scattered points of industrialization ("high-tech"). What kind of industrialization is wholly determined by what levels are needed for jet engine (turbofan, turbojet, air breathing combustion engines, etc.) manufacture.
Knowing this small detail will greatly help me further my worldbuilding process.

$\begingroup$Have you considered the other setting elements necessary for jet engines to be useful, not just technically feasible? Jet engines came into common use as a result of their better performance at high speeds and high altitudes. Even today, low-altitude, lightweight, and short-ranged aircraft typically remain piston-driven.$\endgroup$
– CatgutApr 2 '18 at 17:18

$\begingroup$This is a good answer. It has components I was hoping to get from an answer to such a question; the things I had not thought of. High heat and stress, gears, and machine crafting technology have a logical bondage with one another, where a factor of one motivates a factor in all. Yes, good answer.$\endgroup$
– B.foxApr 1 '18 at 1:25

If you are looking for very simple yet useful(so no Aeolipile) jet engine then look into valveless pulsejets. They are quite noisy, have short lifespan and are quite inefficient(about 700 s against 3000 s for good turbofan) but are extreamly easy to make - you can make one in garage(or this site) like conditions if you have some Bessemer steel sheets and welding, maybe you can make one just by forging wrought iron by hand(nobody tried this one but I think it could work).

They can work on anything that works(even coal dust has been tried). Common misconception about pulsejets is that they do not work at rest - they do, but take-off just on pulsejet would be very inefficient.

For further details I would highly recommend «The enthusiast guide to pulsejet engines» by Simpson Bruce.

Looking at jet engines there were actually a multitude of inventions that needed to occur to overcome:

Weight penalty

Fuel consumption

Safety

Reliability

Sustained operation

Propeller driven engines were well-known throughout the pre-WW1 period, but their limitations were well known, in terms of power and therefore altitude and speed. Ironically, prior to and in the early portion of WW2, the above issues were solved by both german and allied researchers such that the first stable jet engines were able to go into production just prior to the end of the war.

However, these early models during the war were still experimenting with turbofans, with new forms of liquid fuel and new metallurgical techniques. Safety was a major issue, with failures in jet engines quite common. Also at this time they had very limited range.

It would take decades to refine all of the above to give confidence to not just militaries, but to also commercial operators, to allow mass production.

Keep in mind the engine is only one part of the equation necessary for flight. For instance in the Korean War, it was discovered that the US jets were being shot down a lot of the time due to them not having swept wings, where their MIG counterparts did (for serendipitous reasons). Fuel and range was a major issue too until more efficient Bypass turbofans were invented and studies in aerodynamics were perfected. Metal stress and fatigue for multiple flights (even things such as the dangerous discovery regarding square windows) evolved the form of aircraft, and even maintenance - this is why you see a lot of pod engines mounted underwing now, as you can access them to maintain them whereas tri-jets and rear pods were harder to maintain because they were higher up.

Commercial air-craft design is fascinating - they evolve through thousands of small improvements over time to form what they are now.